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Tutorial

Libigl Tutorials¶

Libigl is an open source C++ library for geometry processing research and development. The python bindings combine the rapid prototyping familiar to Matlab to Python programmers with the performance and versatility of C++. The tutorial is a self-contained, hands-on introduction to libigl in Python. Via interactive, step-by-step examples, we demonstrate how to accomplish common geometry processing tasks such as computation of differential quantities and operators, real-time deformation, parametrization, numerical optimization and remeshing. Each section of the lecture notes contains a simple Python example application.

Chapter 0¶

We introduce libigl with a series of self-contained examples. The purpose of each example is to showcase a feature of libigl while applying to a practical problem in geometry processing. In this chapter, we will present the basic concepts of libigl.

Libigl design principles¶

Before getting into the examples, we summarize the two main design principles in libigl:

No complex data types. We mostly use numpy or scipy matrices and vectors. This greatly favors code reusability and interoperability and forces the function authors to expose all the parameters used by the algorithm.
Function encapsulation. Every function is contained in a unique Python function.

Downloading Libigl¶

Libigl can be downloaded from Conda forge:

conda install -c conda-forge igl

All of libigl functionality depends only on numpy and scipy. For the visualization in this tutorial we use meshplot which can be easily installed from Conda:

conda install -c conda-forge meshplot

To start using libigl (with the plots) you just need to import it together with the numpy, scipy, and meshplot.

import igl
import scipy as sp
import numpy as np
from meshplot import plot, subplot, interact

Mesh representation¶

Libigl uses numpy to encode vectors and matrices and scipy for sparse matrices.

A triangular mesh is encoded as a pair of matrices:

v: np.array
f: np.array

v is a #N by 3 matrix which stores the coordinates of the vertices. Each row stores the coordinate of a vertex, with its x, y and z coordinates in the first, second and third column, respectively. The matrix f stores the triangle connectivity: each line of f denotes a triangle whose 3 vertices are represented as indices pointing to rows of f.

A simple mesh made of 2 triangles and 4 vertices.

V = np.array([
    [0., 0, 0],
    [1, 0, 0],
    [1, 1, 1],
    [2, 1, 0]
])

F = np.array([
    [0, 1, 2],
    [1, 3, 2]
])

plot(V, F)

Note that the order of the vertex indices in f determines the orientation of the triangles and it should thus be consistent for the entire surface. This simple representation has many advantages:

It is memory efficient and cache friendly
The use of indices instead of pointers greatly simplifies debugging
The data can be trivially copied and serialized

Libigl provides input and output functions to read and write many common mesh formats. The IO functions are igl.read_* and igl.write_*.

Reading a mesh from a file requires a single libigl function call:

## Load a mesh in OFF format
v, f = igl.read_triangle_mesh("data/bunny.off")

## Print the vertices and faces matrices 
print("Vertices: ", len(v))
print("Faces: ", len(f))

Vertices:  3485
Faces:  6966

The function reads the mesh bumpy.off and returns the v and f matrices. Similarly, a mesh can be written to an OBJ file using:

## Save the mesh in OBJ format
ret = igl.write_triangle_mesh("data/bunny.obj", v, f)

Chapter 1: Discrete Geometric Quantities and Operators¶

This chapter illustrates a few discrete quantities that libigl can compute on a mesh and the libigl functions that construct popular discrete differential geometry operators. It also provides an introduction to basic drawing and coloring routines of our viewer.

Gaussian curvature¶

Gaussian curvature on a continuous surface is defined as the product of the principal curvatures:

$k_G = k_1 k_2.$

As an intrinsic measure, it depends on the metric and not the surface’s embedding.

Intuitively, Gaussian curvature tells how locally spherical or elliptic the surface is ( $k_G>0$ ), how locally saddle-shaped or hyperbolic the surface is ( $k_G<0$ ), or how locally cylindrical or parabolic ( $k_G=0$ ) the surface is.

In the discrete setting, one definition for a “discrete Gaussian curvature” on a triangle mesh is via a vertex’s angular deficit:

$k_G(v_i) = 2π - \sum\limits_{j\in N(i)}θ_{ij},$

where $N(i)$ are the triangles incident on vertex $i$ and $θ_{ij}$ is the angle at vertex $i$ in triangle $j$ (Meyer, 2003).

Just like the continuous analog, our discrete Gaussian curvature reveals elliptic, hyperbolic and parabolic vertices on the domain.

Let’s compute Gaussian curvature and visualize it in pseudocolor. First, calculate the curvature with libigl and then plot it in pseudocolors.

v, f = igl.read_triangle_mesh("data/bumpy.off")
k = igl.gaussian_curvature(v, f)
plot(v, f, k)

Next, compute the massmatrix and divide the gaussian curvature values by area to get the integral average.

m = igl.massmatrix(v, f, igl.MASSMATRIX_TYPE_VORONOI)
minv = sp.sparse.diags(1 / m.diagonal())
kn = minv.dot(k)
plot(v, f, kn)

Curvature directions¶

The two principal curvatures $(k_1,k_2)$ at a point on a surface measure how much the surface bends in different directions. The directions of maximum and minimum (signed) bending are called principal directions and are always orthogonal.

Mean curvature is defined as the average of principal curvatures:

$H = \frac{1}{2}(k_1 + k_2).$

One way to extract mean curvature is by examining the Laplace-Beltrami operator applied to the surface positions. The result is a so-called mean-curvature normal:

$-\Delta \mathbf{x} = H \mathbf{n}.$

It is easy to compute this on a discrete triangle mesh in libigl using the cotangent Laplace-Beltrami operator (Meyer, 2003).

l = igl.cotmatrix(v, f)
m = igl.massmatrix(v, f, igl.MASSMATRIX_TYPE_VORONOI)

minv = sp.sparse.diags(1 / m.diagonal())

hn = -minv.dot(l.dot(v))
h = np.linalg.norm(hn, axis=1)
plot(v, f, h)

Combined with the angle defect definition of discrete Gaussian curvature, one can define principal curvatures and use least squares fitting to find directions (Meyer, 2003).

Alternatively, a robust method for determining principal curvatures is via quadric fitting (Panozzo, 2010). In the neighborhood around every vertex, a best-fit quadric is found and principal curvature values and directions are analytically computed on this quadric.

v1, v2, k1, k2 = igl.principal_curvature(v, f)
h2 = 0.5 * (k1 + k2)
p = plot(v, f, h2, shading={"wireframe": False}, return_plot=True)

avg = igl.avg_edge_length(v, f) / 2.0
p.add_lines(v + v1 * avg, v - v1 * avg, shading={"line_color": "red"})
p.add_lines(v + v2 * avg, v - v2 * avg, shading={"line_color": "green"});

Gradient¶

Scalar functions on a surface can be discretized as a piecewise linear function with values defined at each mesh vertex:

$f(\mathbf{x}) \approx \sum\limits_{i=1}^n \phi_i(\mathbf{x})\, f_i,$

where $\phi_i$ is a piecewise linear hat function defined by the mesh so that for each triangle $\phi_i$ is the linear function which is one only at vertex $i$ and zero at the other corners.

$Hat function \phi_i\phi_i is one at vertex ii, zero at all other vertices, and linear on incident triangles.$

Thus gradients of such piecewise linear functions are simply sums of gradients of the hat functions:

$\nabla f(\mathbf{x}) \approx \nabla \sum\limits_{i=1}^n \phi_i(\mathbf{x})\, f_i = \sum\limits_{i=1}^n \nabla \phi_i(\mathbf{x})\, f_i.$

This reveals that the gradient is a linear function of the vector of $f_i$ values. Because the $\phi_i$ are linear in each triangle, their gradients are constant in each triangle. Thus our discrete gradient operator can be written as a matrix multiplication taking vertex values to triangle values:

$\nabla f \approx \mathbf{G}\,\mathbf{f},$

where $\mathbf{f}$ is $n\times 1$ and $\mathbf{G}$ is an $md\times n$ sparse matrix. This matrix $\mathbf{G}$ can be derived geometrically (Jacobson, 2013).

Libigl’s grad function computes $\mathbf{G}$ for triangle and tetrahedral meshes. Let’s see how this works. First load a mesh and a corresponding surface function. Next, compute the gradient operator g (#F*3 x #V) on the triangle mesh, apply it to the surface function and extract the magnitude.

v, f = igl.read_triangle_mesh("data/cheburashka.off")
u = igl.read_dmat("data/cheburashka-scalar.dmat")

g = igl.grad(v, f)
gu = g.dot(u).reshape(f.shape, order="F")

gu_mag = np.linalg.norm(gu, axis=1)
p = plot(v, f, u, shading={"wireframe":False}, return_plot=True)

max_size = igl.avg_edge_length(v, f) / np.mean(gu_mag)
bc = igl.barycenter(v, f)
bcn = bc + max_size * gu
p.add_lines(bc, bcn, shading={"line_color": "black"});

Laplacian¶

The discrete Laplacian is an essential geometry processing tool. Many interpretations and flavors of the Laplace and Laplace-Beltrami operator exist.

In open Euclidean space, the Laplace operator is the usual divergence of gradient (or equivalently the Laplacian of a function is the trace of its Hessian):

$\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.$

The Laplace-Beltrami operator generalizes this to surfaces.

When considering piecewise-linear functions on a triangle mesh, a discrete Laplacian may be derived in a variety of ways. The most popular in geometry processing is the so-called “cotangent Laplacian” $\mathbf{L}$ , arising simultaneously from FEM, DEC and applying divergence theorem to vertex one-rings. As a linear operator taking vertex values to vertex values, the Laplacian $\mathbf{L}$ is a $n\times n$ matrix with elements:

$L_{ij} = \begin{cases}j \in N(i) &\cot \alpha_{ij} + \cot \beta_{ij},\\ j \notin N(i) & 0,\\ i = j & -\sum\limits_{k\neq i} L_{ik}, \end{cases}$

where $N(i)$ are the vertices adjacent to (neighboring) vertex $i$ , and $\alpha_{ij},\beta_{ij}$ are the angles opposite to edge ${ij}$ .

Libigl implements discrete “cotangent Laplacians” for triangles meshes and tetrahedral meshes, building both with fast geometric rules rather than “by the book” FEM construction which involves many (small) matrix inversions (Sharf, 2007).

First, load a triangle mesh and then calculate the Laplace-Beltrami operator, visualize the normals as pseudocolors.

from scipy.sparse.linalg import spsolve

v, f = igl.read_triangle_mesh("data/cow.off")
l = igl.cotmatrix(v, f)

n = igl.per_vertex_normals(v, f)*0.5+0.5
c = np.linalg.norm(n, axis=1)
p = plot(v, f, c, shading={"wireframe": False}, return_plot=True)

vs = [v]
cs = [c]
for i in range(10):
    m = igl.massmatrix(v, f, igl.MASSMATRIX_TYPE_BARYCENTRIC)
    s = (m - 0.001 * l)
    b = m.dot(v)
    v = spsolve(s, m.dot(v))
    n = igl.per_vertex_normals(v, f)*0.5+0.5
    c = np.linalg.norm(n, axis=1)
    vs.append(v)
    cs.append(c)

@interact(level=(0, 9))
def mcf(level=0):
    p.update_object(vertices=vs[level], colors=cs[level])

interactive(children=(IntSlider(value=0, description='level', max=9), Output()), _dom_classes=('widget-interac…

The operator applied to mesh vertex positions amounts to smoothing by flowing the surface along the mean curvature normal direction. Note that this is equivalent to minimizing surface area. The following example computes conformalized mean curvature flow using the cotangent Laplacian (Kazhdan, 2012)

Mass matrix¶

The mass matrix $\mathbf{M}$ is another $n \times n$ matrix which takes vertex values to vertex values. From an FEM point of view, it is a discretization of the inner-product: it accounts for the area around each vertex. Consequently, $\mathbf{M}$ is often a diagonal matrix, such that $M_{ii}$ is the barycentric or voronoi area around vertex $i$ in the mesh (Meyer, 2003). The inverse of this matrix is also very useful as it transforms integrated quantities into point-wise quantities, e.g.:

$\Delta f \approx \mathbf{M}^{-1} \mathbf{L} \mathbf{f}.$

In general, when encountering squared quantities integrated over the surface, the mass matrix will be used as the discretization of the inner product when sampling function values at vertices:

$\int_S x\, y\ dA \approx \mathbf{x}^T\mathbf{M}\,\mathbf{y}.$

An alternative mass matrix $\mathbf{T}$ is a $md \times md$ matrix which takes triangle vector values to triangle vector values. This matrix represents an inner-product accounting for the area associated with each triangle (i.e. the triangles true area).

Alternative construction of Laplacian¶

An alternative construction of the discrete cotangent Laplacian is by “squaring” the discrete gradient operator. This may be derived by applying Green’s identity (ignoring boundary conditions for the moment):

$\int_S \|\nabla f\|^2 dA = \int_S f \Delta f dA$

Or in matrix form which is immediately translatable to code:

$\mathbf{f}^T \mathbf{G}^T \mathbf{T} \mathbf{G} \mathbf{f} = \mathbf{f}^T \mathbf{M} \mathbf{M}^{-1} \mathbf{L} \mathbf{f} = \mathbf{f}^T \mathbf{L} \mathbf{f}.$

So we have that $\mathbf{L} = \mathbf{G}^T \mathbf{T} \mathbf{G}$ . This also hints that we may consider $\mathbf{G}^T$ as a discrete divergence operator, since the Laplacian is the divergence of the gradient. Naturally, $\mathbf{G}^T$ is a $n \times md$ sparse matrix which takes vector values stored at triangle faces to scalar divergence values at vertices.

v, f = igl.read_triangle_mesh("data/cow.off")
l = igl.cotmatrix(v, f)
g = igl.grad(v, f)

d_area = igl.doublearea(v, f)
t = sp.sparse.diags(np.hstack([d_area, d_area, d_area]) * 0.5)

k = -g.T.dot(t).dot(g)
print("|k-l|: %s"%sp.sparse.linalg.norm(k-l))

|k-l|: 6.654117928693559e-13

Exact Discrete Geodesic Distances¶

The discrete geodesic distance between two points is the length of the shortest path between then restricted to the surface. For triangle meshes, such a path is made of a set of segments which can be either edges of the mesh or crossing a triangle.

Libigl includes a wrapper for the exact geodesic algorithm (Mitchell, 1987) developed by Danil Kirsanov (https://code.google.com/archive/p/geodesic/), exposing it through an Eigen-based API. The function

d = igl.exact_geodesic(v, f, vs, fs, vt, ft)

computes the closest geodesic distances of each vertex in vt or face in ft, from the source vertices vs or faces fs of the input mesh v, f. The output is written in the vector d, which lists first the distances for the vertices in vt, and then for the faces in ft.

v, f = igl.read_triangle_mesh("data/armadillo.obj")

## Select a vertex from which the distances should be calculated
vs = np.array([0])
##All vertices are the targets
vt = np.arange(v.shape[0])

d = igl.exact_geodesic(v, f, vs, vt)#, fs, ft)

strip_size = 0.1
##The function should be 1 on each integer coordinate
c = np.abs(np.sin((d / strip_size * np.pi)))
plot(v, f, c, shading={"wireframe": False})

Chapter 2: Matrices and Linear Algebra¶

Laplace equation¶

A common linear system in geometry processing is the Laplace equation:

$∆z = 0$

subject to some boundary conditions, for example Dirichlet boundary conditions (fixed value):

$\left.z\right|_{\partial{S}} = z_{bc}$

In the discrete setting, the linear system can be written as:

$\mathbf{L} \mathbf{z} = \mathbf{0}$

where $\mathbf{L}$ is the $n \times n$ discrete Laplacian and $\mathbf{z}$ is a vector of per-vertex values. Most of $\mathbf{z}$ correspond to interior vertices and are unknown, but some of $\mathbf{z}$ represent values at boundary vertices. Their values are known so we may move their corresponding terms to the right-hand side.

Conceptually, this is very easy if we have sorted $\mathbf{z}$ so that interior vertices come first and then boundary vertices:

$\left(\begin{array}{cc} \mathbf{L}_{in,in} & \mathbf{L}_{in,b}\\ \mathbf{L}_{b,in} & \mathbf{L}_{b,b}\end{array}\right) \left(\begin{array}{c} \mathbf{z}_{in}\\ \mathbf{z}_{b}\end{array}\right) = \left(\begin{array}{c} \mathbf{0}_{in}\\ \mathbf{z}_{bc}\end{array}\right)$

The bottom block of equations is no longer meaningful so we’ll only consider the top block:

$\left(\begin{array}{cc} \mathbf{L}_{in,in} & \mathbf{L}_{in,b}\end{array}\right) \left(\begin{array}{c} \mathbf{z}_{in}\\ \mathbf{z}_{b}\end{array}\right) = \mathbf{0}_{in}$

We can move the known values to the right-hand side:

$\mathbf{L}_{in,in} \mathbf{z}_{in} = - \mathbf{L}_{in,b} \mathbf{z}_{b}$

Finally we can solve this equation for the unknown values at interior vertices $\mathbf{z}_{in}$ .

However, our vertices will often not be sorted in this way. One option would be to sort V, then proceed as above and then unsort the solution Z to match V. However, this solution is not very general.

With array slicing no explicit sort is needed. Instead we can slice-out submatrix blocks ( $\mathbf{L}_{in,in}$ , $\mathbf{L}_{in,b}$ , etc.) and follow the linear algebra above directly. Then we can slice the solution into the rows of Z corresponding to the interior vertices.

from scipy.sparse.linalg import spsolve

v, f = igl.read_triangle_mesh("data/camelhead.off")

## Find boundary vertices
e = igl.boundary_facets(f)
v_b = np.unique(e)

## List of all vertex indices
v_all = np.arange(v.shape[0])

## List of interior indices
v_in = np.setdiff1d(v_all, v_b)

## Construct and slice up Laplacian
l = igl.cotmatrix(v, f)
l_ii = l[v_in, :]
l_ii = l_ii[:, v_in]

l_ib = l[v_in, :]
l_ib = l_ib[:, v_b]

## Dirichlet boundary conditions from z-coordinate
z = v[:, 2]
bc = z[v_b]

## Solve PDE
z_in = spsolve(-l_ii, l_ib.dot(bc))

plot(v, f, z)

Quadratic energy minimization¶

The same Laplace equation may be equivalently derived by minimizing Dirichlet energy subject to the same boundary conditions:

$\mathop{\text{minimize }}_z \frac{1}{2}\int\limits_S \|\nabla z\|^2 dA$

On our discrete mesh, recall that this becomes

$\mathop{\text{minimize }}_\mathbf{z} \frac{1}{2}\mathbf{z}^T \mathbf{G}^T \mathbf{D} \mathbf{G} \mathbf{z} \rightarrow \mathop{\text{minimize }}_\mathbf{z} \mathbf{z}^T \mathbf{L} \mathbf{z}$

The general problem of minimizing some energy over a mesh subject to fixed value boundary conditions is so wide spread that libigl has a dedicated api for solving such systems.

Let us consider a general quadratic minimization problem subject to different common constraints:

$\mathop{\text{minimize }}_\mathbf{z} \frac{1}{2}\mathbf{z}^T \mathbf{Q} \mathbf{z} + \mathbf{z}^T \mathbf{B} + \text{constant},$

subject to

$\mathbf{z}_b = \mathbf{z}_{bc} \text{ and } \mathbf{A}_{eq} \mathbf{z} = \mathbf{B}_{eq},$

where

$\mathbf{Q}$ is a (usually sparse) $n \times n$ positive semi-definite matrix of quadratic coefficients (Hessian),
$\mathbf{B}$ is a $n \times 1$ vector of linear coefficients,
$\mathbf{z}_b$ is a $|b| \times 1$ portion of $\mathbf{z}$ corresponding to boundary or fixed vertices,
$\mathbf{z}_{bc}$ is a $|b| \times 1$ vector of known values corresponding to $\mathbf{z}_b$ ,
$\mathbf{A}_{eq}$ is a (usually sparse) $m \times n$ matrix of linear equality constraint coefficients (one row per constraint), and
$\mathbf{B}_{eq}$ is a $m \times 1$ vector of linear equality constraint right-hand side values.

This specification is overly general as we could write $\mathbf{z}_b = \mathbf{z}_{bc}$ as rows of $\mathbf{A}_{eq} \mathbf{z} = \mathbf{B}_{eq}$ , but these fixed value constraints appear so often that they merit a dedicated function: min_quad_with_fixed.

Linear equality constraints¶

We saw above that min_quad_with_fixed in libigl provides a compact way to solve general quadratic programs. Let’s consider another example, this time with active linear equality constraints. Specifically let’s solve the bi-Laplace equation or equivalently minimize the Laplace energy:

$\Delta^2 z = 0 \leftrightarrow \mathop{\text{minimize }}\limits_z \frac{1}{2} \int\limits_S (\Delta z)^2 dA$

subject to fixed value constraints and a linear equality constraint:

$z_{a} = 1, z_{b} = -1$ and $z_{c} = z_{d}$ .

Notice that we can rewrite the last constraint in the familiar form from above:

$z_{c} - z_{d} = 0.$

Now we can assembly Aeq as a $1 \times n$ sparse matrix with a coefficient $1$ in the column corresponding to vertex $c$ and a $-1$ at $d$ . The right-hand side Beq is simply zero.

Internally, min_quad_with_fixed solves using the Lagrange Multiplier method. This method adds additional variables for each linear constraint (in general a $m \times 1$ vector of variables $\lambda$ ) and then solves the saddle problem:

$\mathop{\text{find saddle }}_{\mathbf{z},\lambda}\, \frac{1}{2}\mathbf{z}^T \mathbf{Q} \mathbf{z} + \mathbf{z}^T \mathbf{B} + \text{constant} + \lambda^T\left(\mathbf{A}_{eq} \mathbf{z} - \mathbf{B}_{eq}\right)$

This can be rewritten in a more familiar form by stacking $\mathbf{z}$ and $\lambda$ into one $(m+n) \times 1$ vector of unknowns:

$\mathop{\text{find saddle }}_{\mathbf{z},\lambda}\, \frac{1}{2} \left( \mathbf{z}^T \lambda^T \right) \left( \begin{array}{cc} \mathbf{Q} & \mathbf{A}_{eq}^T\\ \mathbf{A}_{eq} & 0 \end{array} \right) \left( \begin{array}{c} \mathbf{z}\\ \lambda \end{array} \right) + \left( \mathbf{z}^T \lambda^T \right) \left( \begin{array}{c} \mathbf{B}\\ -\mathbf{B}_{eq} \end{array} \right) + \text{constant}$

Differentiating with respect to $\left( \mathbf{z}^T \lambda^T \right)$ reveals a linear system and we can solve for $\mathbf{z}$ and $\lambda$ . The only difference from the straight quadratic minimization system, is that this saddle problem system will not be positive definite. Thus, we must use a different factorization technique (LDLT rather than LLT): libigl’s min_quad_with_fixed automatically chooses the correct solver in the presence of linear equality constraints.

The following example first solves with just fixed value constraints (left: 1 and -1 on the left hand and foot respectively), then solves with an additional linear equality constraint (right: points on right hand and foot constrained to be equal).

v, f = igl.read_triangle_mesh("data/cheburashka.off")

## Two fixed points: Left hand, left foot should have values 1 and -1
b = np.array([4331, 5957])
bc = np.array([1., -1.])
B = np.zeros((v.shape[0], 1))

## Construct Laplacian and mass matrix
L = igl.cotmatrix(v, f)
M = igl.massmatrix(v, f, igl.MASSMATRIX_TYPE_VORONOI)
Minv = sp.sparse.diags(1 / M.diagonal())

## Bi-Laplacian
Q = L @ (Minv @ L)

## Solve with only equality constraints
Aeq = sp.sparse.csc_matrix((0, 0))
Beq = np.array([])
_, z1 = igl.min_quad_with_fixed(Q, B, b, bc, Aeq, Beq, True)

## Solve with equality and linear constraints
Aeq = sp.sparse.csc_matrix((1, v.shape[0]))
Aeq[0,6074] = 1
Aeq[0, 6523] = -1
Beq = np.array([0.])
_, z2 = igl.min_quad_with_fixed(Q, B, b, bc, Aeq, Beq, True)

## Normalize colors to same range
min_z = min(np.min(z1), np.min(z2))
max_z = max(np.max(z1), np.max(z2))
z = [(z1 - min_z) / (max_z - min_z), (z2 - min_z) / (max_z - min_z)]

## Plot the functions
p = plot(v, f, z1, shading={"wireframe":False}, return_plot=True)

@interact(function=[('z0', 0), ('z1', 1)])
def sf(function):
    p.update_object(colors=z[function])

/Users/teseo/conda/envs/base2/lib/python3.6/site-packages/ipykernel_launcher.py:23: SparseEfficiencyWarning: Changing the sparsity structure of a csc_matrix is expensive. lil_matrix is more efficient.
/Users/teseo/conda/envs/base2/lib/python3.6/site-packages/ipykernel_launcher.py:24: SparseEfficiencyWarning: Changing the sparsity structure of a csc_matrix is expensive. lil_matrix is more efficient.



interactive(children=(Dropdown(description='function', options=(('z0', 0), ('z1', 1)), value=0), Output()), _d…

Eigen Decomposition¶

Libigl has rudimentary support for extracting eigen pairs of a generalized eigen value problem:

$Ax = \lambda B x$

where $A$ is a sparse symmetric matrix and $B$ is a sparse positive definite matrix. Most commonly in geometry processing, we let $A=L$ the cotangent Laplacian and $B=M$ the per-vertex mass matrix (Vallet, 2008). Typically applications will make use of the low frequency eigen modes. Analogous to the Fourier decomposition, a function $f$ on a surface can be represented via its spectral decomposition of the eigen modes of the Laplace-Beltrami:

$f = \sum\limits_{i=1}^\infty a_i \phi_i$

where each $\phi_i$ is an eigen function satisfying: $\Delta \phi_i = \lambda_i \phi_i$ and $a_i$ are scalar coefficients. For a discrete triangle mesh, a completely analogous decomposition exists, albeit with finite sum:

$\mathbf{f} = \sum\limits_{i=1}^n a_i \phi_i$

where now a column vector of values at vertices $\mathbf{f} \in \mathcal{R}^n$ specifies a piecewise linear function and $\phi_i \in \mathcal{R}^n$ is an eigen vector satisfying:

$\mathbf{L} \phi_i = \lambda_i \mathbf{M} \phi_i$ .

Note that Vallet & Levy (Vallet, 2008) propose solving a symmetrized standard eigen problem $\mathbf{M}^{-1/2}\mathbf{L}\mathbf{M}^{-1/2} \phi_i = \lambda_i \phi_i$ . Libigl implements a generalized eigen problem solver so this unnecessary symmetrization can be avoided.

Often the sum above is truncated to the first $k$ eigen vectors. If the low frequency modes are chosen, i.e. those corresponding to small $\lambda_i$ values, then this truncation effectively regularizes $\mathbf{f}$ to smooth, slowly changing functions over the mesh (Hildebrandt, 2011). Modal analysis and model subspaces have been used frequently in real-time deformation (Barbic, 2005).

In the following example, the first k eigen vectors of the discrete Laplace-Beltrami operator are computed and displayed in pseudocolors atop the beetle. Low frequency eigen vectors of the discrete Laplace-Beltrami operator vary smoothly and slowly over the model. At first, calculate the Laplace-Betrami operator and solve the generalized Eigen problem with scipy/arpack. Then, rescale the Eigen vectors and visualize them.

v, f = igl.read_triangle_mesh("data/beetle.off")
l = -igl.cotmatrix(v, f)
m = igl.massmatrix(v, f, igl.MASSMATRIX_TYPE_VORONOI)

k = 10
d, u = sp.sparse.linalg.eigsh(l, k, m, sigma=0, which="LM")

u = (u - np.min(u)) / (np.max(u) - np.min(u))
bbd = 0.5 * np.linalg.norm(np.max(v, axis=0) - np.min(v, axis=0))

p = plot(v, f, bbd * u[:, 0], shading={"wireframe":False, "flat": False}, return_plot=True)

@interact(ev=[("EV %i"%i, i) for i in range(k)])
def sf(ev):
    p.update_object(colors=u[:, ev])

interactive(children=(Dropdown(description='ev', options=(('EV 0', 0), ('EV 1', 1), ('EV 2', 2), ('EV 3', 3), …

Chapter 3: Shape deformation¶

Modern mesh-based shape deformation methods satisfy user deformation constraints at handles (selected vertices or regions on the mesh) and propagate these handle deformations to the rest of the shape smoothly and without removing or distorting details. Libigl provides implementations of a variety of state-of-the-art deformation techniques, ranging from quadratic mesh-based energy minimizers, to skinning methods, to non-linear elasticity-inspired techniques.

Biharmonic deformation¶

The period of research between 2000 and 2010 produced a collection of techniques that cast the problem of handle-based shape deformation as a quadratic energy minimization problem or equivalently the solution to a linear partial differential equation.

There are many flavors of these techniques, but a prototypical subset are those that consider solutions to the bi-Laplace equation, that is a biharmonic function (Botsch, 2004). This fourth-order PDE provides sufficient flexibility in boundary conditions to ensure $C^1$ continuity at handle constraints in the limit under refinement (Jacobson, 2010).

Biharmonic surfaces¶

Let us first begin our discussion of biharmonic deformation, by considering biharmonic surfaces. We will casually define biharmonic surfaces as surface whose position functions are biharmonic with respect to some initial parameterization:

$\Delta^2 \mathbf{x}' = 0$

and subject to some handle constraints, conceptualized as “boundary conditions”:

$\mathbf{x}'_{b} = \mathbf{x}_{bc}.$

where $\mathbf{x}'$ is the unknown 3D position of a point on the surface. So we are asking that the bi-Laplacian of each of spatial coordinate function to be zero.

In libigl, one can solve a biharmonic problem with harmonic_weights and setting $k=2$ (bi-harmonic).

This produces a smooth surface that interpolates the handle constraints, but all original details on the surface will be smoothed away. Most obviously, if the original surface is not already biharmonic, then giving all handles the identity deformation (keeping them at their rest positions) will not reproduce the original surface. Rather, the result will be the biharmonic surface that does interpolate those handle positions.

Thus, we may conclude that this is not an intuitive technique for shape deformation.

Biharmonic deformation fields¶

Now we know that one useful property for a deformation technique is “rest pose reproduction”: applying no deformation to the handles should apply no deformation to the shape.

To guarantee this by construction we can work with deformation fields (ie. displacements) $\mathbf{d}$ rather than directly with positions $\mathbf{x}$ . Then the deformed positions can be recovered as

$\mathbf{x}' = \mathbf{x}+\mathbf{d}.$

A smooth deformation field $\mathbf{d}$ which interpolates the deformation fields of the handle constraints will impose a smooth deformed shape $\mathbf{x}'$ . Naturally, we consider biharmonic deformation fields:

$\Delta^2 \mathbf{d} = 0$

subject to the same handle constraints, but rewritten in terms of their implied deformation field at the boundary (handles).

$\mathbf{d}_b = \mathbf{x}_{bc} - \mathbf{x}_b.$

Again we can use harmonic_weights with $k=2$ , but this time solve for the deformation field and then recover the deformed positions:

v, f = igl.read_triangle_mesh("data/decimated-max.obj")
v[:,[0, 2]] = v[:,[2, 0]] # Swap X and Z axes
u = v.copy()

s = igl.read_dmat("data/decimated-max-selection.dmat")
b = np.array([[t[0] for t in [(i, s[i]) for i in range(0, v.shape[0])] if t[1] >= 0]]).T

## Boundary conditions directly on deformed positions
u_bc = np.zeros((b.shape[0], v.shape[1]))
v_bc = np.zeros((b.shape[0], v.shape[1]))

for bi in range(b.shape[0]):
    v_bc[bi] = v[b[bi]]

    if s[b[bi]] == 0: # Don't move handle 0
        u_bc[bi] = v[b[bi]]
    elif s[b[bi]] == 1: # Move handle 1 down
        u_bc[bi] = v[b[bi]] + np.array([[0, -50, 0]])
    else: # Move other handles forward
        u_bc[bi] = v[b[bi]] + np.array([[-25, 0, 0]])

p = plot(v, f, s, shading={"wireframe": False, "colormap": "tab10"}, return_plot=True)

@interact(deformation_field=True, step=(0.0, 2.0))
def update(deformation_field, step=0.0):
    # Determine boundary conditions
    u_bc_anim = v_bc + step * (u_bc - v_bc)

    if deformation_field:
        d_bc = u_bc_anim - v_bc
        d = igl.harmonic_weights(v, f, b, d_bc, 2)
        u = v + d
    else:
        u = igl.harmonic_weights(v, f, b, u_bc_anim, 2)
    p.update_object(vertices=u)

interactive(children=(Checkbox(value=True, description='deformation_field'), FloatSlider(value=0.0, descriptio…

Relationship to “differential coordinates” and Laplacian surface editing¶

Biharmonic functions (whether positions or displacements) are solutions to the bi-Laplace equation, but also minimizers of the “Laplacian energy”. For example, for displacements $\mathbf{d}$ , the energy reads

$\int\limits_S \|\Delta \mathbf{d}\|^2 dA,$

where we define $\Delta \mathbf{d}$ to simply apply the Laplacian coordinate-wise.

By linearity of the Laplace(-Beltrami) operator we can reexpress this energy in terms of the original positions $\mathbf{x}$ and the unknown positions $\mathbf{x}' = \mathbf{x} - \mathbf{d}$ :

$\int\limits_S \|\Delta (\mathbf{x}' - \mathbf{x})\|^2 dA = \int\limits_S \|\Delta \mathbf{x}' - \Delta \mathbf{x})\|^2 dA.$

In the early work of Sorkine et al., the quantities $\Delta \mathbf{x}'$ and $\Delta \mathbf{x}$ were dubbed “differential coordinates” (Sorkine, 2004). Their deformations (without linearized rotations) is thus equivalent to biharmonic deformation fields.

Polyharmonic deformation¶

We can generalize biharmonic deformation by considering different powers of the Laplacian, resulting in a series of PDEs of the form:

$\Delta^k \mathbf{d} = 0.$

with $k\in{1,2,3,\dots}$ . The choice of $k$ determines the level of continuity at the handles. In particular, $k=1$ implies $C^0$ at the boundary, $k=2$ implies $C^1$ , $k=3$ implies $C^2$ and in general $k$ implies $C^{k-1}$ .

The following example deforms a flat domain (left) into a bump as a solution to various $k$ -harmonic PDEs.

v, f = igl.read_triangle_mesh("data/bump-domain.obj")
u = v.copy()

# Find boundary vertices outside annulus
vrn = np.linalg.norm(v, axis = 1)
is_outer = [vrn[i] - 1.00 > -1e-15 for i in range(v.shape[0])]
is_inner = [vrn[i] - 0.15 < 1e-15 for i in range(v.shape[0])]
in_b = [is_outer[i] or is_inner[i] for i in range(len(is_outer))]

b = np.array([i for i in range(v.shape[0]) if (in_b[i])]).T
bc = np.zeros(b.size)

for bi in range(b.size):
    bc[bi] = 0.0 if is_outer[b[bi]] else 1.0

c = np.array(is_outer)

p = plot(u, f, c, shading={"wire_width": 0.01, "colormap": "tab10"}, return_plot=True)                

@interact(z_max=(0.0, 1.0), k=(1, 4))
def update(z_max, k):
    z = igl.harmonic_weights(v, f, b, bc, int(k))
    u[:, 2] = z_max * z
    p.update_object(vertices=u)

interactive(children=(FloatSlider(value=0.5, description='z_max', max=1.0), IntSlider(value=2, description='k'…

Chapter 4: Parametrization¶

In computer graphics, we denote as surface parametrization a map from the surface to $\mathbf{R}^2$. It is usually encoded by a new set of 2D coordinates for each vertex of the mesh (and possibly also by a new set of faces in one to one correspondence with the faces of the original surface). Note that this definition is the inverse of the classical differential geometry definition.

A parametrization has many applications, ranging from texture mapping to surface remeshing. Many algorithms have been proposed, and they can be broadly divided in four families:

Single patch, fixed boundary: these algorithm can parametrize a disk-like part of the surface given fixed 2D positions for its boundary. These algorithms are efficient and simple, but they usually produce high-distortion maps due to the fixed boundary.
Single patch, free boundary: these algorithms let the boundary deform freely, greatly reducing the map distortion. Care should be taken to prevent the border to self-intersect.
Global parametrization: these algorithms work on meshes with arbitrary genus. They initially cut the mesh in multiple patches that can be separately parametrized. The generated maps are discontinuous on the cuts (often referred as seams).
Global seamless parametrization: these are global parametrization algorithm that hides the seams, making the parametrization “continuous”, under specific assumptions that we will discuss later.

Harmonic parametrization¶

Harmonic parametrization (Eck, 2005) is a single patch, fixed boundary parametrization algorithm that computes the 2D coordinates of the flattened mesh as two harmonic functions.

The algorithm is divided in 3 steps:

Detection of the boundary vertices
Map the boundary vertices to a circle
Compute two harmonic functions (one for u and one for the v coordinate). The harmonic functions use the fixed vertices on the circle as boundary constraints.

The algorithm is coded with libigl in the following example. bnd contains the indices of the boundary vertices, bnd_uv their position on the UV plane, and “1” denotes that we want to compute an harmonic function (2 will be for biharmonic, 3 for triharmonic, etc.). Note that each of the three functions is designed to be reusable in other parametrization algorithms. The UV coordinates are then used to apply a procedural checkerboard texture to the mesh.

v, f  = igl.read_triangle_mesh("data/camelhead.off")
## Find the open boundary
bnd = igl.boundary_loop(f)

## Map the boundary to a circle, preserving edge proportions
bnd_uv = igl.map_vertices_to_circle(v, bnd)

## Harmonic parametrization for the internal vertices
uv = igl.harmonic_weights(v, f, bnd, bnd_uv, 1)
v_p = np.hstack([uv, np.zeros((uv.shape[0],1))])

p = plot(v, f, uv=uv, shading={"wireframe": False, "flat": False}, return_plot=True)

@interact(mode=['3D','2D'])
def switch(mode):
    if mode == "3D":
        plot(v, f, uv=uv, shading={"wireframe": False, "flat": False}, plot=p)
    if mode == "2D":
        plot(v_p, f, uv=uv, shading={"wireframe": True, "flat": False}, plot=p)

interactive(children=(Dropdown(description='mode', options=('3D', '2D'), value='3D'), Output()), _dom_classes=…

Least squares conformal maps¶

Least squares conformal maps parametrization (Levy, 2002) minimizes the conformal (angular) distortion of the parametrization. Differently from harmonic parametrization, it does not need to have a fixed boundary.

LSCM minimizes the following energy:

\[ E_{LSCM}(\mathbf{u},\mathbf{v}) = \int_X \frac{1}{2}| \nabla \mathbf{u}^{\perp} - \nabla \mathbf{v} |^2 dA \]

which can be rewritten in matrix form as (Mullen, 2008):

\[ E_{LSCM}(\mathbf{u},\mathbf{v}) = \frac{1}{2} [\mathbf{u},\mathbf{v}]^t (L_c - 2A) [\mathbf{u},\mathbf{v}] \]

where $L_c$ is the cotangent Laplacian matrix and $A$ is a matrix such that $[\mathbf{u},\mathbf{v}]^t A [\mathbf{u},\mathbf{v}]$ is equal to the vector area of the mesh.

Using libigl, this matrix energy can be written in a few lines of code. The cotangent matrix can be computed using igl.cotmatrix. Note that we want to apply the Laplacian matrix to the u and v coordinates at the same time, thus we need to extend it taking the left Kronecker product with a 2x2 identity matrix. The area matrix is computed with igl.vector_area_matrix.

The final energy matrix is $L_{flat} - 2A$ . Note that in this case we do not need to fix the boundary. To remove the null space of the energy and make the minimum unique, it is sufficient to fix two arbitrary vertices to two arbitrary positions. The full source code is provided in the following LSCM parametrization example.

v, f = igl.read_triangle_mesh("data/camelhead.off")

# Fix two points on the boundary
b = np.array([2, 1])

bnd = igl.boundary_loop(f)
b[0] = bnd[0]
b[1] = bnd[int(bnd.size / 2)]

bc = np.array([[0.0, 0.0], [1.0, 0.0]])

# LSCM parametrization
_, uv = igl.lscm(v, f, b, bc)

p = plot(v, f, uv=uv, shading={"wireframe": False, "flat": False}, return_plot=True)

@interact(mode=['3D','2D'])
def switch(mode):
    if mode == "3D":
        plot(v, f, uv=uv, shading={"wireframe": False, "flat": False}, plot=p)
    if mode == "2D":
        plot(uv, f, uv=uv, shading={"wireframe": True, "flat": False}, plot=p)

interactive(children=(Dropdown(description='mode', options=('3D', '2D'), value='3D'), Output()), _dom_classes=…

Chapter 5: External libraries¶

An additional positive side effect of using matrices as basic types is that it is easy to exchange data between libigl and other software and libraries.

Triangulation of closed polygons¶

The generation of high-quality triangle and tetrahedral meshes is a very common task in geometry processing. We provide wrappers in libigl to triangle and Tetgen.

A triangle mesh with a given boundary can be created as in the following example. e is a set of boundary edges (#e by 2), h is a set of 2D positions of points contained in holes of the triangulation (#h by 2) and (v,f) is the generated triangulation. Additional parameters can be passed to triangle, to control the quality: "a0.005q" enforces a bound on the maximal area of the triangles and a minimal angle of 20 degrees. In the following example, the interior of a square (excluded a smaller square in its interior) is triangulated.

## Input polygon
v = np.array([[-1.0, -1.0], [1, -1], [1, 1], [-1, 1], [-2, -2], [2, -2], [2, 2], [-2, 2]])
e = np.array([[0, 1], [1, 2], [2, 3], [3, 0], [4, 5], [5, 6], [6, 7], [7, 4]])
h = np.array([[0.0, 0.0]])

v, f = igl.triangulate(v, e, h, flags="a0.05q")
ext = np.zeros((v.shape[0],1))
v = np.append(v, ext, axis=1)
plot(v, f)

Tetrahedralization of closed surfaces¶

Similarly, the interior of a closed manifold surface can be tetrahedralized using the function tetrahedralize which wraps the Tetgen library.

v, f = igl.read_triangle_mesh("data/fertility.off")

## Tetrahedralize the interior
success, tv, tt, tf = igl.tetrahedralize(v, f, "pq1.414Y")

## Compute barycenters
b = igl.barycenter(tv, tt)

p = plot(v, f, return_plot=True)

@interact(level=(1,9))
def step(level):
    t = level / 9.0
    v = b[:, 2] - np.min(b[:, 2])
    v /= np.max(v)

    s = []
    for i in range(v.shape[0]):
        if v[i] < t:
            s.append(i)

    f_tmp = np.zeros((len(s) * 4, 3), dtype="int32")

    for i in range(len(s)):
        tet = tt[s[i]]
        f_tmp[i*4:i*4+4] = np.array([[tet[1], tet[0], tet[2]], [tet[0], tet[1], tet[3]],
                                     [tet[1], tet[2], tet[3]], [tet[2], tet[0], tet[3]]],dtype="int32")

    plot(tv, f_tmp, plot=p)

interactive(children=(IntSlider(value=5, description='level', max=9, min=1), Output()), _dom_classes=('widget-…

Baking ambient occlusion¶

Ambient occlusion is a rendering technique used to calculate the exposure of each point in a surface to ambient lighting. It is usually encoded as a scalar (normalized between 0 and 1) associated with the vertice of a mesh.

Formally, ambient occlusion is defined as:

\[ A_p = \frac{1}{\pi} \int_\omega V_{p,\omega}(n \cdot \omega) d\omega \]

where $V_{p,\omega}$ is the visibility function at p, defined to be zero if p is occluded in the direction $\omega$ and one otherwise, and $d\omega$ is the infinitesimal solid angle step of the integration variable $\omega$ .

The integral is usually approximated by casting rays in random directions around each vertex. This approximation can be computed using the function:

ao = igl.ambient_occlusion(v, f, v_samples, n_samples, 500)

that given a scene described in v and f, computes the ambient occlusion of the points in v_samples whose associated normals are n_samples. The number of casted rays can be controlled (usually at least 300-500 rays are required to get a smooth result) and the result is returned in ao, as a single scalar for each sample.

Ambient occlusion can be used to darken the surface colors, as shown in the following example:

v, f = igl.read_triangle_mesh("data/fertility.off")

n = igl.per_vertex_normals(v, f)

# Compute ambient occlusion factor using embree
ao = igl.ambient_occlusion(v, f, v, n, 50)
ao = 1.0 - ao

plot(v, f, ao, shading={"colormap": "gist_gray"})

Chapter 6: Miscellaneous¶

Libigl contains a wide variety of geometry processing tools and functions for dealing with meshes and the linear algebra related to them: far too many to discuss in this introductory tutorial. We’ve pulled out a couple of the interesting functions in this chapter to highlight.

Mesh Statistics¶

Libigl contains various mesh statistics, including face angles, face areas and the detection of singular vertices, which are vertices with more or less than 6 neighbours in triangulations or 4 in quadrangulations.

The example computes these quantities and does a basic statistic analysis that allows to estimate the isometry and regularity of a mesh:

v, f = igl.read_triangle_mesh("data/horse_quad.obj")

## Count the number of irregular vertices, the border is ignored
irregular = igl.is_irregular_vertex(v, f) 
v_count = v.shape[0]
irregular_v_count = np.sum(irregular)
irregular_ratio = irregular_v_count / v_count

print("Irregular vertices: \n%d/%d (%.2f%%)\n"%(irregular_v_count, v_count, irregular_ratio * 100))

## Compute areas, min, max and standard deviation
area = igl.doublearea(v, f) / 2.0

area_avg = np.mean(area)
area_min = np.min(area) / area_avg
area_max = np.max(area) / area_avg
area_ns = (area - area_avg) / area_avg
area_sigma = np.sqrt(np.mean(np.square(area_ns)))

print("Areas (Min/Max)/Avg_Area Sigma: \n%.2f/%.2f (%.2f)\n"%(area_min, area_max, area_sigma))

## Compute per face angles, min, max and standard deviation
angles = igl.internal_angles(v, f)
angles = 360.0 * (angles / (2 * np.pi))

angle_avg = np.mean(angles)
angle_min = np.min(angles)
angle_max = np.max(angles)
angle_ns = angles - angle_avg
angle_sigma = np.sqrt(np.mean(np.square(angle_ns)))

print("Angles in degrees (Min/Max) Sigma: \n%.2f/%.2f (%.2f)\n"%(angle_min, angle_max, angle_sigma))

Irregular vertices: 
659/2400 (27.46%)

Areas (Min/Max)/Avg_Area Sigma: 
0.01/6.26 (0.88)

Angles in degrees (Min/Max) Sigma: 
2.36/171.79 (25.02)

The first row contains the number and percentage of irregular vertices, which is particularly important for quadrilateral meshes when they are used to define subdivision surfaces: every singular point will result in a point of the surface that is only C^1.

The second row reports the area of the minimal element, maximal element and the standard deviation. These numbers are normalized by the mean area, so in the example above 5.33 max area means that the biggest face is 5 times larger than the average face. An ideal isotropic mesh would have both min and max area close to 1.

The third row measures the face angles, which should be close to 60 degrees (90 for quads) in a perfectly regular triangulation. For FEM purposes, the closer the angles are to 60 degrees the more stable will the optimization be. In this case, it is clear that the mesh is of bad quality and it will probably result in artifacts if used for solving PDEs.

References¶

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